The present invention relates to measurement of the shape of a workpiece and, more particularly, to a method for calibrating a shape measuring system with a variable distance between the scanning probe and the measuring point of the workpiece.
A layout of a typical shape measuring system, such as a coordinate measuring machine (CMM), is illustrated in FIG. 1. A workpiece 10 with a measuring point 11 on the surface of workpiece 10, is set on a working table 21 of a shape measuring system 20. A scanning probe 41 senses measuring point 11, by either touching measuring point 11 in a contact shape measurement, or by pointing toward measuring point 11 and measuring a distance D between measuring point 11 and a scanning probe datum point 43 in a non-contact shape measurement. A scanning probe holder 42 (usually a probe head), as part of a scanning probe assembly 40, holds scanning probe 41 onto a motion system platform 30. Motion system platform 30 is movable in three axes (X, Y, Z) of a fixed coordinate frame 22 of shape measuring system 20. The motion and position of motion system platform 30 are monitored according to the coordinates of a reference point 31 on motion system platform 30. A vector r.sub.CMM, with Cartesian components (X.sub.CMM, Y.sub.CMM, Z.sub.CMM) in coordinate frame 22, represents the position of reference point 31 relative to a fixed origin point 23 of coordinate frame 22, whereas a vector r, with Cartesian components (X, Y, Z) in coordinate frame 22, represents the coordinate of measuring point 11 relative to origin point 23. The coordinate r.sub.CMM of reference point 31, is read whenever the scanning probe senses a measuring point on the surface of the workpiece.
However, because of a spatial offset between reference point 31 and measuring point 11, the coordinate r of measuring point 11 is displaced relative to the measured position r.sub.CMM of reference point 31, according to the formula: r=r.sub.CMM +.delta., where a displacement vector .delta.=.DELTA.+D is composed of two components: a vector .DELTA., with Cartesian components (.DELTA..sub.X, .DELTA..sub.Y, .DELTA..sub.Z) in coordinate frame 22, which is a constant offset between scanning probe datum point 43 and reference point 31, and D=D.multidot.i, which is a distance vector between measuring point 11 and scanning probe datum point 43, where i is a unit vector pointing from the scanning probe datum point 43 toward the measuring point.
It is worthwhile to mention, that in the case of present art contact scanning probes, the distance D between the scanning probe datum point and the measuring points of the workpiece is constant. It is then common to define the scanning probe datum point as the touching point of the scanning probe with the surface of the workpiece. In such a case, D=0, and hence .delta.=.DELTA..
Thus, in the case of shape measurement using either a present art contact scanning probe, or a non-contact scanning probe with a constant distance D between the scanning probe datum point and the measuring points of the workpiece, if all points of the workpiece are measured with the same scanning probe configuration, then the displacement .delta. between reference point 31 and measuring point 11 is the same for all measuring points of the workpiece. Hence, measuring the coordinate r.sub.CMM of reference point 31 is sufficient in this case for determining the shape of the workpiece.
The situation is more complicated when several different scanning probe configurations are used for measuring the same workpiece, using either a present art contact scanning probe, or a non-contact scanning probe with a constant distance D between the scanning probe datum point and the measuring points of the workpiece. The various scanning probe configurations can differ by the offset .DELTA. between the scanning probe datum point 43 and the reference point 31, and/or the distance D between the scanning probe datum point 43 and the measuring point 11, and/or the inclination of the scanning probe. An example of changing the scanning probe configuration, is by attaching an extension, such as a metal shaft, to a contact scanning probe to make it longer, when necessary for measuring hard-to-reach parts of the workpiece.
Thus, in such cases where several different scanning probe configurations are used for measuring the same workpiece, the displacement .delta. between reference point 31 and measuring point 11, is not the same for all measuring points of the workpiece. Hence, measuring the coordinate r.sub.CMM of reference point 31 is not sufficient in this case for determining the shape of the workpiece. Thus, appropriate prior art calibration methods exist, which provide the data that is necessary for deriving the coordinate r measuring point 11 from the measured position coordinate r.sub.CMM of reference point 31.
The basic idea of these prior art calibration methods which are suitable for shape measurement using either a present art contact scanning probe, or a non-contact scanning probe with a constant distance D between the scanning probe datum point and the measuring points of the workpiece, is to utilize a calibration object with a known geometry and a particular calibration point.
A vector diagram of a setup of a shape measuring system for a prior art calibration process is shown in FIG. 2. A calibration object 50 with a known geometry, and including a calibration point 55, is fixed by a fixture 59 to working table 21 of shape measuring system 20. Calibration object 50 is usually a calibration sphere with a known radius R, and the center point of the calibration sphere serving as calibration point 55.
For each scanning probe configuration individually, several measuring points on the circumference of calibration sphere 50 are scanned, and the corresponding position r.sub.CMM of the reference point on the motion system platform for each measuring point on the calibration sphere, is recorded.
For the sake of describing the prior art calibration methods, a measuring point 51" indicates a measuring point "n" out of a plurality of N+1 measuring points numbered "0", "1", "2", . . . , "N" on the circumference of calibration sphere 50, for the calibration process of a particular scanning probe configuration. Point 31" in FIG. 2 indicates the corresponding position of reference point 31 of FIG. 1 when scanning probe 41 senses measuring point 51".
Referring further to FIG. 2, vector equations between the various positions and distances can be written for each of the plurality of N+1 measuring points. However, for the sake of simplicity, these equations will be explicitly presented for measuring point "n" (51"), as a representative for all N+1 measuring points.
The basic relation is: r(n)=r.sub.CMM (n)+.delta., where a position vector r.sub.CMM (n), with Cartesian components (X.sub.CMM (n), Y.sub.CMM (n), Z.sub.CMM (n)) in coordinate frame 22, represents the measured position of reference point 31" relative to an origin point 23 of coordinate frame 22. .delta. is the constant but unknown displacement vector between measuring point 51" and reference point 31", with Cartesian components (.delta..sub.X, .delta..sub.Y, .delta..sub.Z) in coordinate frame 22. Thus, the coordinate vector r(n), with Cartesian components (X(n), Y(n), Z(n)) in coordinate frame 22, representing the coordinates of measuring point 51" relative to an origin point 23 of coordinate frame 22, can be calculated from the measured position vector r.sub.CMM (n) of reference point 31" when knowing the displacement vector .delta..
In order to provide the data that is needed for calculating the coordinate of a measuring point from the corresponding position of the reference point, the geometrical relation: .vertline.r(n)"r.sub.C.vertline..sup.2 =R.sup.2 is utilized in the prior art calibration methods for each of the plurality of N+1 measuring points on the circumference of calibration sphere 50, where the coordinate vector r.sub.C, with Cartesian components (X.sub.C, Y.sub.C, Z.sub.C) in coordinate frame 22, represents the constant but unknown coordinates of calibration sphere's center point 55 relative to origin point 23 of coordinate frame 22. Inserting the relation for r(n): r(n)=r.sub.CMM (n)+.delta., the geometrical relation .vertline.r(n)-r.sub.C.vertline..sup.2 =R.sup.2, reads: .vertline.r.sub.CMM (n)-(r.sub.C).sub.CMM.vertline..sup.2 =R.sup.2 ; where the coordinate vector (r.sub.C).sub.CMM =r.sub.C -.delta.. represents the effective position of reference point 31 that would have been obtained if scanning probe 41 would have been sensing calibration sphere's center point 55 directly. The Cartesian components ((X.sub.C).sub.CMM, (Y.sub.C).sub.CMM, (Z.sub.C).sub.CMM) of coordinate vector (r.sub.C).sub.CMM, in coordinate frame 22, are: (X.sub.C).sub.CMM =X.sub.C -.delta..sub.X, (Y.sub.C).sub.CMM =Y.sub.C -.delta..sub.Y, and (Z.sub.C).sub.CMM =Z.sub.C -.delta..sub.Z.
Thus, in order to derive the value of the coordinate vector (r.sub.C).sub.CMM for each scanning probe configuration, from the measured coordinate vectors r.sub.CMM (n) for all N+1 measuring points over the circumference of calibration sphere 50, a best fit is performed using a chi-square merit function, .chi..sup.2 ((r.sub.C).sub.CMM), that incorporates the geometrical relation .vertline.r(n)-r.sub.C.vertline..sup.2 =R.sup.2 for the whole set of N+1 measuring points: ##EQU1##
There are well known non-linear least squares methods, such as Levenberg-Marquardt method, for performing this best fit. These methods are described in a variety of text books, such as for example, "Numerical Recipes in C", W. H. Press et al, 2nd Edition, Cambridge University Press, 1992.
The next steps in the prior art calibration methods are as follows. One of the scanning probe configurations is defined as a master scanning probe configuration. Then, the data that is needed for transforming the position of the reference point obtained by scanning the surface of the workpiece with the various probe configurations, to one common basis, is derived. This common basis is the position of the reference point that would have been obtained by scanning the surface of the workpiece by the master scanning probe configuration.
This transformation into a reading of the reference point position that would have been obtained by scanning with the master scanning probe configuration, is based on the following considerations. If r is the coordinate of a measuring point on the surface of the workpiece, then the corresponding position r.sub.CMM of the reference point on the motion system platform when scanning with a particular scanning probe configuration, is: r.sub.CMM =r-.delta., and the relation between the same coordinate r and the position of the reference point r.sub.CMM (M) that would have been obtained with the master scanning configuration, is: r.sub.CMM (M)=r-.delta.(M), where .delta. and .delta.(M) are the displacement between the reference point position and the measuring point, for the particular scanning probe configuration and the master probe configuration, respectively. Thus, the transformation from r.sub.CMM into r.sub.CMM (M) is given by: r.sub.CMM (M)=r.sub.CMM +.delta.-.delta.(M), where the difference .delta.-.delta.(M) can be determined according to the difference between the center of the calibration sphere (r.sub.C).sub.CMM as obtained for the particular scanning probe configuration, and the center of the calibration sphere (r.sub.C).sub.CMM (M) as obtained for the master scanning probe configuration, namely: .delta.-.delta.(M)=(r.sub.C).sub.CMM (M)-(r.sub.C).sub.CMM.
The situation gets even more complicated when the surface of the workpiece is scanned using a scanning probe with a variable distance D between the scanning probe datum point and the measuring point of the workpiece. An example of such a scanning probe, is a non-contact scanning probe, such as the laser-based WIZ probe, manufactured by Nextec of Tirat Hacarmel, Israel. The WIZ probe is suitable for measuring with a high resolution of 0.1 .mu.m, a variable distance D between the scanning probe datum point and a measuring point on the surface of the workpiece, with a nominal value of the distance D of 50 mm, and a non-zero dynamic range of .+-.5 mm.
In such a scanning with a variable distance D, the displacement .delta.=.DELTA.+D between the reference point and the measuring point of the workpiece, varies through the scanning of the workpiece. Thus, the above mentioned prior art calibration methods that are based on deriving the difference between a constant offset .delta. and the constant offset .delta.(M) of a master scanning probe configuration, by determining the coordinates of a calibration point, such as a center of a calibration sphere, are not suitable for scanning with a variable distance D.
There is thus a widely recognized need for, and it would be highly advantageous to have, a calibration method for shape measurement with a variable distance D between the scanning probe datum point and the measuring point of the workpiece, for providing the data that is needed for deriving the coordinate r of a measuring point of the workpiece from the measured corresponding position r.sub.CMM of the reference point on the motion system platform, and the measured distance D between the scanning probe datum point and the measuring point of the workpiece, in an accurate and straightforward manner.